 Proces is in fact about what we call the initiation of motion. And before we do that, you have to know some basics. We already discussed that in the introduction course, but I will briefly rehearse it. Why is erosion or the initiation of motion of interest in dredging? Well, first of all, and I will show something about that. If you have the hopper loading process, at the end of the loading process of a hopper, you get very high velocities of the mixture above the sediment. En die high velocities cause erosion. En die resulten in de fact that the efficiency of the loading process goes to zero. So at the end, you are not loading anything anymore. In fact, all the sand that goes into the hopper leaves the hopper through the overflow. And if you can model the erosion well, you can predict the loading process en you can determine when to stop loading. Another process in dredging is the hydraulic transport. And in hydraulic transport, hydraulic transport is the transport of solids with water in general with the fluid. But in dredging, the fluid is always water. So we transport solids, mainly sand and gravel, through pipelines mixed with water. En if those solids have quite a large particle size, or if the concentration is high, what will happen is that you get a bed, a sediment in the bottom of the pipe. And that means at the interface between the bed and the fluid, you also have erosion. In fact, the bed forming in such a pipeline is always an equilibrium between settling sedimentation on one hand and erosion on the other hand. So you have to be able to understand the settling process of the particles in the pipeline on one hand and the erosion process on the other hand. But you can also be busy with what we call beach nourishment. Beach nourishment is that there's not enough sand on the beach anymore. En usually with hopper dredges, we will dredge sand from the North Sea and pump it on the beach. Could be with a pipeline. And then if you do that and you pump the slurry on the beach, what you want is the sand to settle on the beach and the water to flow back through the sea. But you can imagine that if the water flow is too high, if you have relatively too much water, that will cause erosion on the beach and part of the sand that you are dumping on the beach will flow back to the sea. So it's also interesting to understand that process. So first the settling process. Well, what do we have? We assume a spherical particle. In reality the particles are not sphere spherical, they are angular, but we assume a spherical particle. And that spherical particle is subject basically to two forces. You could split it up in three or four forces, but we normally use two forces. You have an upwards force and a downwards force. En if those two forces are equal, that's where you have what we call the terminal settling velocity. We talk about a terminal settling velocity because if you hold a particle in the water and release it, first it has to accelerate to get to that terminal settling velocity. So it's the final velocity. What are those two forces? We have the upwards force, which is the so-called drag force. Every object which is in a flow, whether it's a fluid flow or an air flow, doesn't matter, every object will have a drag force. We know from cars, cars have a certain drag force when you are driving high speed. En in fact, it's an easy exercise to determine the power required to overcome this drag force if you know the maximum power and the maximum velocity of your car. The drag force, it's a CD and I mentioned cars. In good cars, the CD value would be something like 0.25, 0.3, but if you buy a Hummer, probably it's 1. The resistance could be 4 times as big and also the fuel consumption at high speed. Not at low speed, but at high speed. So that's the CD. We will see what the CD is for particles later. Half rho, in this case the water density, maar als het een andere fluid is, kan je de densiteit van de andere fluid gebruiken. V is de terminale zettingvelocatie, want we kijken naar de eindvelocatie en A is de cross-section van de particle perpendicular naar de velociteit. Oh, dus dat is één force. Dat is de opwereiding, de resisting force, in feite. Dan de opwereiding is de graffitie-force, maar omdat we onder water zijn, moeten we kijken naar de verbodigde graffitie-force. Niet de graffitie-force boven water, maar de graffitie-force onder water. Dus je krijgt de densiteit van de particle, in dit geval. Normaal wordt het een kwart. Minus de densiteit van water in order de verbodigde densiteit te krijgen. Dan g, dan hebben we de weight, de volume en een shapefactor. Waarom een shapefactor? Als we een spheren hebben, dan is de shapefactor één. Maar als je een spheren niet hebt, als je een reale centparticle hebt, dan is het normaal de ratio tussen de diameter en de volume van het particle niet één. Het is kleiner dan één. Voor een reale cent, we gebruiken vaak een facteur 0,7, iets in dat gebied. Dus die zijn de twee voorsetjes. En als je die twee voorsetjes met elkaarzelf maakt, vind je deze equatie, die is de generelle equatie voor de terminale zettingfacetie. En je kunt zien dat je de vierteur G hebt, de verbodigde densiteit, de diameter van de particle, de shapefactor. En dan kun je hier de densiteit van water en de cd-factor. In veel andere papieren gebruik ik ook de vt, de t van termineel. Hier gebruik ik de s van zetting. Boven in de zes kun je het zien. Nou, de cd-value. Als je kunt calculeren over de zettingfacetie, moet je de cd-value van particles weten. En deze graf is voor sphiers. Wat zien we op de horizontale zetting? We hebben de particle Reinhold's nummer. En de particle Reinhold's nummer is de zettingfacetie times de diameter divide by the viscosity. En daar hebben we al een probleem gezien, want de zettingfacetie is in de Rijnhold's nummer, maar we moeten de zettingfacetie om de zettingfacetie te vertellen. Dus het is een implicit situatie. Maar ze hebben al dat lange tijd gegaan. Nou, wat heb je? Als je heel klein particles hebt, dan heb je ook heel klein Reinhold's nummers. En je kunt zien dat er een soort van linair relatie tussen de cd-value en de Reinhold's nummer is. Als ik het correct denk, is het 24 divide by de Reinhold's nummer. Dus de kleinste of de Reinhold's nummer, de groter de cd-value. Dat heeft te doen met het effect dat in dit gebied de flow rond de particle is lemmener. Het is fiscus flow, dus alles is beperkt door fiscus frictie. Dan op heel hoge Reinhold's nummers, in dit gebied, de cd-value is bijna constant. En dat is omdat in dat gebied we de turbulent flow rond de particle hebben. En dat geeft ons een constant cd-value. De cd-value voor sphiers in dat gebied zou behoorlijk 0.445 zijn. Nou, je moet het niet zeggen omdat het 3 decimals is, dus het is heel zeker. Maar dat is in dit gebied. Inbetween hier, natuurlijk, je hebt een transitione gebied waar je een fiscus-influentie en een turbulent-influentie hebt en je een transitione gebied krijgt. Nu, op heel hoogen Reinhold's nummers, je ziet een zudde geluid. En dat zudde geluid is iets wat je krijgt omdat als je de flow rond de particle hebt, als je de flow in general hebt, we zeggen dat de flow verbindt met de surface, dat betekent dat de surface, de velociteit van de flow, altijd 0 is. Dus je krijgt een velociteitprofile die begint op 0 op de surface van het object. Maar op heel hoogen Reinhold's nummers kan er een situatie zijn waar het disconnect wordt waar de flow niet in contact met de surface meer is. Dus de flow zal laten gaan van de surface. En omdat van dat effect zudde de cd-valuid dropt. Ze gebruiken, ik weet niet of ze het nog gebruiken, maar een paar jaar geleden, ze gebruiken dat effect. Dus dit effect in speed skating. Ze zetten die strippen op de arm en op de hoofd en die strippen zetten dat de flow zal laten gaan van je body. Dus het is niet connecten op de manier, maar het gaat straks. Het zal laten gaan. En de theorie is dat, omdat van dat de cd-valuid van de skater reduzigt enorm. En dan kan hij faster skaten. Het is eigenlijk dat, als het werkt, want ik zie het niet meer in speed skating, maar 10 jaar geleden, het werkt niet, maar het werkt. Oké, wat ik begon is dat het meestal werkt op de kleinste deel, want dat is waar je de hoge snelheid hebt. Dus voor de 500 meter en de 1000 meter waar ze echt hoge snelheid hebben, kan ze naar 60 kilometer per uur gaan. Dat is waar dit werkt, maar als je de 10 kilometer doet, het lijkt dat de effect is al veel minder. Maar anyway. Ik denk dat ze ook deze testen voor zwemming. En in fact, dan kun je zeggen, ja, maar in zwemming, de speeds zijn veel minder, die is waar, maar dan in zwemming, je bent in water, en de densiteit van water is ongeveer 1.000 keer hoog dan de densiteit van het air. Dus het zou kunnen zijn dat je in dezelfde regionen krijgt en het werkt. En misschien met de Tour de France, met bicyclerijden, het zou ook werken. Maar ik denk dat je het werkt, maar het is niet gevoeligd, het is niet ver. Oké, oké. Maar in fact, ik weet dat ze veel testen hebben over dit in de Delft Universiteit, want in, about 10 jaar geleden, er was een van de Nationaal skaters, die was een student van de TU Delft, en ze hadden hem alle testen in de windtunnel van de TU Delft, dus dat is waar ze het allemaal testen. En het werkt. Dus, oké. Hier zie je wat equaties om de cd-valuërs te kregen. Je kunt het zien voor heel klein rinalsnummer, 24 divided by the rinalsnummer. En dan kun je zeggen, ja, maar hoe kan ik de rinalsnummer kregen omdat ik niet de zettingvelocatie weet. Nou, we zullen zien, als we dit in de equaties substitueren, dan zullen we een explicite equaties vinden en we het kunnen zetten. De transissie-area is niet echt van interesse, het is gewoon een mathematically equaties om de twee, om de twee regio's te connecten. En dan in de, voor de heel hoogte, de turbulente regio, hoogte rinalsnummer, we vinden 0.445. Maar remember, dit is allemaal voor spier, ja, niet voor reale zandparticles. Nou, dit is eigenlijk dezelfde, maar het stopt voordat je die rinalsnummer hebt en in deze graf, ik heb een aantal equaties bepaald. Mr. Huisman was een professor op civil engineering. En hij deed veel research op dit, dus hij heeft zijn equaties. Dan heb je Terten en Levenspeel. Ze hebben ook een heel goede equaties. In fact, ik prefererde deze equaties omdat het gewoon één equaties is, die is expliceerd en het is heel makkelijk te gebruiken. En als je de kruis comparet, in fact, hier, de rinalsnummer, called iteration and the black one called iteration, is de impliceerd equaties gevoeligd, dus dat is helemaal theoretisch, maar je kunt zien de verschil tussen de vier equaties en dan hier de gevoeligde punten. De verschil is niet te veel, dus in fact, alle equaties doen er heel goed. En als je over de reale rinalsnummer gaat, het probleem is veel meer, wat is de reale shapefactor van de rinalsnummer en de accuratie in dat shapefactor is veel minder dan de accuratie van die equaties. Dit graf zorgt voor particles van veel verschillende shapefactoren. Hier zie je de shapefactoren. De shapefactor begint met 1.0, dus die zijn de spier, die zijn de rinalsnummer die je ziet op de onderkant, en die zijn ongeveer 0.445, maar je kunt zien dat het niet zelfs is. Maar dan gaat het naar een shapefactor van 0.3, die zijn die spier, dus dat is hierover. En dat betekent dat dit 1 is, dit is 2, dus we zijn rond 3, een shapecd-value van about 3. Als je particles met een shapefactor van 0.3 hebt, wat lijkt het dan? Wel, voor eentje, als je een hetsel hebt, je weet het hetsel van in de zee, hetsel van als ze die zet, zullen de mannen van een dier uit de mannen zetten, dus het zal zo zijn. En gewoon throwen we een echte hetsel in de water en je zet dat soms geval iets saveert. Dus de dingen zijn hetselen. Deel is ook belangrijk in de Freshman, want het gebeurt heel vaak dat als je op de zee van neer een beek draagt, de zand is vol van schijnen of al least schijnen fragmenten. Dat betekent dat je die stukken van schijnen in je pipeline krijgt. En wat doen ze doen? Nou, omdat ze zo hoge draakcoefficiënt zijn, zullen ze niet heel makkelijk zetten, want ze gaan zo, en voordat ze naar het einde van de pipe, er is wat turbulenten om ze weer op te gaan. Maar als ze zetten, als ze de bed opbouwen, de cd-value in de andere richting is zo klein dat de vloer niet meer zeer kan worden. Dus ze gaan de bed opbouwen, protek de bed, en dat betekent dat wanneer de dredge master de lijnspeed in de pipeline is, je kan hartelijk die schijnen van de bed removeden meer, zodat je in het trouwtje bent. En vorig jaar denk ik dat we wat researchen hebben gedaan in dit, en we ontdekken dat het meeste schijf fragmenten kunnen worden bepaald tot particles van 2 mm, als je wilt calculeren hoe hoog de spiegel moet worden om ze te erodigen. Dus als ik de lijnspeed die 2 mm erodigd is van 2 mm, zou ik zo een lijnspeed gebruiken om de schijf fragmenten te erodigen, wanneer ze meer kunnen zijn, dat zijn vaak schijf fragmenten veel minder dan 2 mm, maar dat is wat je nodig hebt. Ja, en je ziet veel verschillende factoren tussen. Nou, er zijn ook mensen in de past die eigenlijk de zettingvelocatie van reale schijf fragmenten ontdekten. En dan natuurlijk de resultaat van waar de schijf kwam. Als het van een estuairie is, dan zijn de partijen al vrij rond. Dus de cd-value zal niet te hoog zijn, maar als het van een licht in de montagel zou zijn, dan hebben je heel angelaar partijen, en dat zou een verschillende resultaat geven. Maar anyway, die zijn de equations die je in allerlei literatuur zult vinden. Voor heel geweldige partijen, kleiner dan 0,1 mm, gebruiken we de Stokes-equation en je kunt zien de fact dat het in het begin is gevolgd om die cd-equations te substitueren. En je vindt een relatief simpele equation, maar je kunt zien dat de zettingvelocatie proportioneer is dan de zettingvelocatie van de partijen-diameter. Ja. Dan gaan we niet kijken naar de verandering, dat is dit, maar voor partijen groter dan 1 mm, we vinden de zoveel called Rittenger equation en je kunt zien dat de velociteit proportioneer is dan de zettingvelocatie van de partijen-diameter. In alle die equations, en dat is nog hoe iedereen ze in de literatuur zet, de partijen-diameter is in 1 mm en de velociteit is in 1 mm per seconde, dus ze gebruiken niet de SI-system, maar ik wilde je de originele equations laten zien. Dus als je ze gebruikt, bewaren je ze, het is 1 mm en 1 mm per seconde en in de meeste andere equations, gebruiken we de SI-system, dat betekent dat alles in 1 mm en 1 mm per seconde is en er is een factor 1000 tussen. Dus je moet dat compenseren op een examen, als je die equations gebruikt. Aan de onderdeel heb ik de equation voor de, wat ik geef, de relatieve onderdeelde densiteit. Sommige mensen het het specifiek grafitee of de specifiek densiteit. Maar ik geef het het relatieve onderdeelde densiteit, omdat het relatief 2 water is. Dus het is de weight van partijen onder water. Dus je krijgt de kwartensdensiteit minus de waterdensiteit divideerd door de waterdensiteit. En voor de meeste normaal onderdeelde densiteit zou een waarde ongeveer 1,65. Maar het betekent wat densiteit gebruik je voor water? De meeste tijd gebruiken we 1,0. Maar als je huidige water met een heel hoge huidige content de densiteit kan 1,03 zijn. En als je de 1,03 gebruikt hier en hier, dan de relatieve onderdeelde densiteit zou 1,59 zijn. Want je bent niet divideerd door een meer. Dat zou een klein verschil maken in de uitgang. Dus met examinerende vragen ga ik altijd in account dat een persoon de één gebruikt en een andere de 1,03 gebruikt. En dus er is altijd een klein marge in de uitgang van calculaties. Dit is wat het lijkt. De red lijn is de stokeslijn. Hier, oh, hier zie je de zetting velociteit in millimeters per seconde. Dus ik deed dat omdat dan kan je connecten met de equaties. Dus red is de stokeslijn. Blu is de ritingerlijn. Green is de stokeslijn. En in feite wat ik vond is dat de stokeslijn ongeveer perfect koopt deze gebied voor heel klein partijken, maar ook de gebied van grote partijken. Dus waarom gebruik ik drie equaties als die stokeslijn al alles koopt. Ik heb de vierde equaties van Mr. Tzangke, een German professor. Hij is nog steeds in de stokeslijn. En je kunt zien dat voor de klein partijken het exact dezelfde is. Voor de grote partijken vond hij een beetje hoog stokeslijn velociteit, maar de verschil is niet te veel. Dus je kan dat gebruiken. En de Tzangke equaties, het mooiste ding van het is dat het niet alleen de partijken diameter ook de fiscositie van de fluid. Dus je kunt dat gebruiken voor temperatuur bijvoorbeeld. Want als ik op de stokeslijn waar de water 4 graden is, de fiscositie van de water is veel meer dan als ik op de tropiek waar de water 25 graden is. De verschil zou misschien 50 procent misschien zijn. In fact, right now I'm busy with research on hydraulic transport and I'm searching for data in literature and I found the Ph.D. thesis of a guy from Canada and with each test he mentioned the temperature at which he did the test and the temperatures range from nine degrees up to 25 degrees. So using this fiscositie by using the Tzangke equation while if you would use the Stokes equation you only have the diameter to play with. You cannot change other things. So that's why I prefer the Tzangke equation. Here I did some calculations just to show you the difference with different shape factors. So here you can see three cases with 0.7 which would be normal sand and then here you see some cases with 0.5 0.5 means a much more angular sand and you can see there is a difference between the two in the graph it doesn't seem the difference is too high but if for example if you take this point at one one millimeter per second and I go to the other shape factor here I would have 15 or something like that which is a difference of 50%. So you can see the shape factor really matters. This one is not important now. One more thing about settling and that's hindered settling. In many applications in civil engineering the concentrations of the solids in the fluid are not too high if you have normal erosion and you look at the sediment transport through a river concentrations are not too high maybe civil engineers disagree with me but in our application it's low concentrations. At low concentrations the settling velocity is not influenced too much by the concentration. It's influenced but not too much. But we go up to concentrations of let's say 50%. So in dredging nowadays they really with those big hopper dredges they really dredge densities of 1.6, 1.7 and then you have concentrations, volumetric concentrations between 40 and 50%. That's much higher dan normally what you will find in rivers. Well what happens if you have a 40% concentration and the particles start settling that means the volume first of all the volume of the particles that are settling which is a volume moving down the same volume of water has to move up which means the relative the settling velocity is the relative velocity between the water and the particles. So if I have an upwards flow velocity of the water the absolute settling velocity will be reduced. So that's one effect. But you also have another effect that if a particle is settling and there's a particle behind it then the particle behind it will be influenced by the velocity of the particle in front. That's why in the Tour de France all those bicycle riders they ride behind each other when they are going at very high speed because the first one should catch the wind and then everybody will go behind it. So you have a number of effects in high concentrations on one hand the upwards velocity of the water on the other hand the fact that particles can be in the shadow of other particles and the total effect is that the settling velocity will be reduced as a function of the concentration. Long time ago there were two researchers in fact you had Mr. Richardson and Mr. Tsarky here I mentioned Richardson from what I remember Richardson was the student of Mr. Tsarky but because Richardson did the research in his master thesis he was the first author of the publication and if you look at the scientific impact then the score of Mr. Richardson and Tsarky is very low. Why is that? Well, the way they determine the score is that it is good for you as a PhD to know the way they determine the score is they count the number of references to each paper and then your score is so suppose you have ten papers and the tenth paper has ten references and the first nine have much more then your score is a ten but in the way they calculate the here's index so they look at at which publication is the number of references the same so if you have ten papers the tenth paper has ten references your score is a ten so those people they just have one paper with six thousand references in the way they calculate the here's index their score is one because they just have one paper so they have a here's index of one but they have officially six thousand references but then officially maybe one hundred thousand because also in all our student reports everybody who is busy with this is referring to those people but you will not find it in the statistics I'm a little bit critical on the system you can understand so what does it look like well, Vs was the terminal settling velocity of one particle Vc is the terminal settling velocity if you have a certain concentration and the ratio between the two is one minus the volumetric concentration to a power of beta this beta you can see in this graph so for very small Reynolds numbers the beta is I think normally we use this line it's four point six nine I thought and then for high Reynolds numbers it's two point three two point four with this equation you can determine the beta if you know the Reynolds number of the particle so it's quite a high power that means if you would have a concentration let's say 0.4 then you get 0.6 to the power of something like four well that becomes a very small number and that means with 40% concentration there's not much left of the settling velocity then you can also imagine that if you are loading a hopper dredge and you do that with a very high concentration the scent is hardly settling anymore because the concentration is so high and you have to take that into account if you want to predict how much time do I need to fill up my hopper dredge ok then we go to erosion well I already told you one of the applications is the hopper dredge because we like to know what happens at the end of the loading process I assume everybody already knows how a hopper dredge works I don't assume everybody has been on a hopper dredge but maybe later that will happen and let's take a look at the loading process I hope well in the pdf I will put on blackboard you can study this graph a little bit better but what we normally do at the horizontal axis you have time here you have zero zero is when we actually start dredging but to get all the information from the graph we always start the graph at the point where the previous cycle stopped dredging so where the hopper is completely full with sand and we start sailing so that's here in this case minus 250 minutes we start where the ship is full and we start sailing then the first thing that happens is that there is still some water above the sediment because when we stop dredging when we turn off the pumps you have sediment in the hopper with a layer of water on top well we want to get rid of that water because that water is weight causes draught of the ship and the more draught you have the more resistance the ship will have sailing back to the burrow area or the dump area so we want to get rid of that's this small line then we get phase 2 which is just sailing to the dump place then you get phase 3 phase 3 is dumping so emptying the ship and phase 3 can be in 3 ways I already explained I think in the introduction course you can use your bottom doors or valves so in the bottom of the ship you have doors, valves you open them and by gravity the material will flow out usually we also use some water jets to make it easier and faster but the other 2 ways are that you fluidize the sand in the hopper so you don't open the bottom doors you just fluidize the material at the bottom of the hopper it will go into a pipeline to the pump and then with the pump you can either use rainbowing or you use a floating pipeline well in both cases the unloading will be about will cost about the same time as the loading so this is a picture where you use the bottom doors which takes maybe 5 minutes it's a matter of minutes but if you would fluidize the load in the hopper and pump it ashore it could take an hour it completely depends on the hopper and the sand et cetera et cetera but it takes much longer ok here in phase 4 the hopper is empty but in practice it's not completely empty because if the bottom doors were open then what will happen is that the level of the water inside the hopper is the same as outside that's the way it works so when you close those bottom doors there's water in the hopper so in fact this graph is not 100% correct because there is water in the hopper now in some cases the contractor will say I will remove that water because I want to start with an empty hopper nowadays there are also cases where they say it's better to start with a hopper full of water but at this point you don't want the water because it causes draft of the ship so you don't like it you sail back to what we call the borough area so that's the area where you are actually dredging and there you start dredging I will not explain the different types of hoppers because you have two ways of loading of a hopper but you start loading up to a point where you reach the overflow level until you reach the overflow level everything will stay in the hopper water and sand because you don't reach the overflow so the mixture cannot go anywhere so it stays in the hopper here you reach the overflow so what happens after you reach the overflow every time when one cubic meter of mixture goes into the hopper one cubic meter of water should leave the hopper through the overflow because the volume is limited so sand is replacing water so that's why the steepness of this curve is much less because you are replacing water with sand then at this point we reach the point where the total load in the hopper is at the maximum so the total weight is at the maximum this is what we call a constant tonnage loading system after we reach the maximum weight in the hopper automatically the overflow is lowered by hydraulic cylinders in order to keep the total weight in the hopper constant so every time when I replace water by sand the weight is increasing because the weight of the sand is 2.65 and the weight of the water is 1.0 so I have to lower my overflow otherwise the weight would increase and the ship would sink and we don't want that so this is the region where you have a constant tonnage in your hopper by lowering the overflow but now we talk about the total weight in the hopper if we look at the effective weight the effective weight is the weight of the sand and you can talk about effective weight in two ways you can say okay I will count the sediment including the pore water or I am only counting the weight of the solids and that is two ways of counting well this graph is the sediment including the pore water and this graph, this line is just the weight of the solids and if you know the porosity of the sand it's just a factor, it's simple now what do we do so here you see the weight of the sediment and at the end somewhere here you get erosion and you get a curve if you would continue the graph it would become horizontal at the point where everything that goes into the hopper leaves the hopper because the erosion is so high now when should we stop loading well what is production production is the total amount of solids whether you include the pore water or not doesn't matter it's the total amount of solids divided by time and then you get either cubic meters per second or you get tons per second whatever you like so what do I want I want my production to be at a maximum because if I have maximum production I earn the most money and basically contractors are not in the world just to transport sand they also want to make money they want to have a profit so if I make a line from my starting point to the curve the point where that line is a tangent a raak line in Dutch, a tangent with the curve, with the loading curve that's the point where I have maximum production because if I divide the load by the time this is the load, this is the time where this angle is at a maximum I have my maximum production right? so that's what I do I draw this line I see where it just touches the loading line well that's in this point I go down and I see that I should stop after about 75 minutes because if I would load longer you can see the curve is becoming horizontal here and that means this angle will decrease again so my production will also decrease so I should stop at that point now the problem in real dredging of course this is theoretical but if you don't understand the theoretical case how can you find something for real life the problem is that you measure all those signals with a certain accuracy how to measure the load in your hopper how to measure it well how they do it if you put pressure sensors in the bottom of your ship then you can measure the draft at that point of your ship so if you do that at a number of points in the bottom of your ship you can measure the bottom line of the ship if you know the whole shape of your ship so from the shipyard who built it you get we call it the Karane diagram which shows you the whole shape of the ship then if you know a number of points at the bottom of the ship you can calculate the water displacement of the ship because you know exactly which part is underwater ok, so if I can calculate the water displacement of the ship I know the total weight of the ship it's the ship including all the sand and everything I can also do that before I start dredging and that way I measure the weight of the empty ship then if I deduct those two things from each other I have the weight of the load but you should realize this means I take the difference of two very big numbers so what is the accuracy but there is no other way because I cannot put a 50.000 cubic meter hopper on a weighing device in the ship that's impossible so this is the only way to do it now I also need the volume of the material in the hopper so what do they have they have those acoustic transducers above the hopper and with those acoustic transducers they measure the water surface in the hopper and because if you are loading the hopper the sand will not settle equally over the full surface because it depends on where is the inlet where is the overflow so the ship may also have some trim so I need to measure at a number of spots what is the surface of the water but if I know it and again in combination with the design of the whole shape of the hopper I can calculate the volume of material in the hopper so then I have my volume and I have my weight no, no, no, no but the thing is if I know but Kase will explain that in the next lecture because he is talking about hopper loading but if you know the weight in the hopper and you know the total volume in the hopper you can calculate how much sediment is in the hopper you need those two numbers in order to calculate so in order to calculate this curve because what I measure is this curve and what I measure is the volume in the hopper those two numbers I can measure and then based on that I can construct this curve or this curve and those two but now I have something we call the law of Bernoulli everybody should be familiar with that so that means if I measure the pressure at the bottom of the hopper and the ship is sailing I have pressure plus half rho v squared is a constant so if I have velocity the pressure is lower so I measure a lower pressure so I think there is less material in the hopper so when to measure it also happens for example in Eimeiden there is a lot of sand and gravel dredging in the North Sea and then those ships come into the port but at the entrance of the channel in Eimeiden there is an undeep area so if the ship sails over that undeep area suddenly the layer of water under the ship is much less which means the velocities under the ship will be much higher and every time when you pass that it looks like there is less material in the hopper and everybody knows it's not true but if you want to have an accurate measurement in fact the ship should stop should not have speed at all but they want to go on because it will cost time if you stop so it's still difficult to do it accurately last year I had a case in a company where they had measurements we were comparing the measurements with the theory and I found some strange things and then I analyzed the measurements and it appeared one of the acoustic transducer broke down so they measured the volume of the hopper just based on one correct transducer and one wrong transducer and then the signal in your computer goes anywhere and you get very but if they would still use those signals either you get paid too less or too much but something is wrong ok we will have a break what it looks like at the end of the loading cycle so you have a hopper full of sediment this is the inlet but in reality the inlet will look differently because an inlet like this would cause a lot of turbulence and you don't want that but anyway this is what it could look like and then you get a high flow of the mixture above the sediment and if those flow velocities are too high nothing will settle anymore in fact whether it's erosion or whether the material just will not settle anymore that's a complicated question probably it's an equilibrium where still some particles will settle but at the same time the same amount of particles will erode you can model such a thing in a very simple way so first I will show you the simple way you assume a number of particles on top of the sediment you have a flow above it and that flow will result in a shear stress on top of the particles the particles have a certain weight and at the bottom those particles have a friction coefficient with the rest of the sediment which results in a friction force F and at the moment where a friction stress you it's also a shear stress at the moment where the driving shear stress of the fluid is equal to the resisting shear stress of the friction the particles will start moving and that's what we call initiation of motion you should distinguish between initiation of motion and erosion because initiation of motion determines when particles will start to move it will not give you an answer on how many particles are moving when we talk about erosion we always talk about the flux of particles that are being transported so in kilograms per second or tons per second or whatever but erosion is the amount of material that's moving initiation of motion is the moment where it starts moving so those are two different things what we will discuss is initiation of motion when do particles start to move well if you look at this problem then we get an equation where the S is the velocity the D is the diameter of the particle and this equation gives you the velocity where particles of that diameter will start moving in the equation you find the friction coefficient of course because we have friction at the bottom the porosity of the sediment and this lambda is the friction coefficient at the top of the bed based on the so called moody diagram and there are many equations for that to determine the lambda usually that lambda will have a value of 0.01 0.02 it depends on the particle size in fact if we use normal values for friction coefficient lambda etc you will get this equation which looks a little bit like the equation for the settling velocity it's quite similar you can also reverse this equation and then you get this equation and what is the use of that well the use of that is that if I know the velocity above the bed I can calculate the diameter of a particle that will just start moving and normally when you are loading a hopper you know the velocity above the bed because that's what you can calculate and then I can calculate which particles will start moving and then I assume that particles that start moving won't settle anymore History the whole theory about initiation of motion started in 36 when an American Mr. Shields did research in a German laboratory and Mr. Shields he found that if you are below a certain curve you will not have particles that are moving if you are above that curve particles start moving and Mr. Shields he made a graph like this and you can see the points that he measured on the horizontal axis of this graph you have the Reynolds number not exactly of the particle but the Reynolds number of the boundary flow and vertically he made another dimensionless parameter which nowadays we call the Shields parameter but in fact that Shields parameter is the dimensionless shear stress later I will show you how you can determine those things it's not too complicated so what do we see we see this well in fact it's not a line it has a certain bandwidth a certain thickness en dat's because you can never say exactly if I'm below a certain point I have movement of the particle and above that line I don't there is a certain bandwidth inside that bandwidth particles could start moving but that's usually in science most of the time you will find a bandwidth and you can say there's a probability that if I'm inside that bandwidth it starts moving in this case then after Mr. Shields other people continued with this research so basically what they do they have what we call a flume which normally is a rectangular cross section very long you put sediment at the bottom and then you have a certain flow velocity over the bed and you can increase that flow velocity and at the moment where particles start moving you say hey I have another point for my graph and then you can decide if it enough if I have one particle that starts moving or do I need 10 per square meter or 100 per square meter and in fact different researchers have different criteria and that's also why it's difficult to compare all the research from the past because people may have had different criteria for what do they call initiation of motion but anyway this is a graph of somebody else and those black points as far as I recall are the points of Mr. Shields and the other points are from other people so basically again if you are below this bandwidth so in this area there should not be initiation of motion if you are above particles start moving still you can see points here that are below where it also started moving well one of the things and we will see later is that the CD value plays an important role and what did people do most researchers in the past did research with spheres and a sphere has a relatively low drag coefficient we already saw that in one of the graphs then if suddenly you do research with real send and real send has a much higher drag coefficient that means the resisting force of the pressure force of the fluid on the particle is much bigger so if you use real send or gravel the whole curve should go down should be lower because it's easier for particles to start moving here you can see again the points of Mr. Shields and a fit curve there are some fit curves that you can find in literature I often use Soulsby and White House for that but there are many more equations with some equations you can see that for very small particles you have a horizontal asymptotic behavior while in the original theory of Mr. Shields he taught here you have a line under a certain angle moving up and the whole point is that the particles in that area are so small that the question is can I still talk about particles and I did some research on that myself because if the particles become so small you should consider send is chemically inert and send particles are so big that van der Waals forces do not play a role but if I go all the way here I'm talking of particles of one micron and if I have particles of one micron also if they are quartz so they are chemically inert but one micron quartz particles are subject to van der Waals forces and that means suddenly those forces they are attractive forces so those very tiny particles will start sticking together that means it's much more difficult to make them start moving because either they behave like a bigger particle because they stick together or the attractive forces pulling them downwards is so big that the water can hardly make them erode so you could say Mr. Shields is right and it goes up because of those van der Waals forces but if you do not count the van der Waals forces you probably get a horizontal asymptotic behavior well here you have a lot of measurements of different researchers at the bottom you can find which researchers are used and I didn't put any curve in this graph just to give you a feeling what does it look like well what you can see is this could start with an asymptotic behavior and go down but it could also go straight up if you look at the cloud of points you also see that there is quite some bandwidth but what you can see is that in this area there is supposed to be a sort of minimum and in this area it looks like it's horizontal again but this is what you get as a researcher I digitized all those points from literature so all the points are real measurements but I also found that for example those yellow points they were from real sand so they have a higher drag coefficient while the other colors were measurements of spheres so it's not strange that you see many of the yellow points lower than the other points this is another way of looking at it Mr. Jilström also in 36 and you will see that more often in science that at the same moment in different places in the world people start finding the same kind of results why? because at that moment science is ready for such a thing so Mr. Jilström is from Scandinavia he did his research independent from Mr. Schilz and Mr. Jilström made a graph which is easier to use but it also has a disadvantage the advantage of the Schilz curve is that because he used dimensionless parameters on both axes so you can put all the measurements in one graph because it's dimensionless the advantage of Mr. Jilström is that on the axis here you see the grain size and here you see the flow velocity above the material so if you know your particle size the flow velocity immediately you can see where do I get erosion basically this is the Jilström curve so this is where you get erosion and if you try to convert the Schilz curve to the Jilström curve this part is roughly the same this part is not but if you look at it he says erosion of consolidated mud mud consolidated could be something like clay and you can imagine if you have a flow above clay it's much more difficult to get erosion than when you have a sand like material I found later that if I do the full Schilz curve for sand and I project it then you get a line that continues going down but apparently Jilström did his research in this area for sand and this area for mud, clay like or silt so what are you comparing this is the same Jilström curve but I made some fit equations to be able to calculate the whole curve and also with Jilström you have a certain bandwidth so the top is the green line the bottom is the red line and that's roughly the bandwidth that you have in the Jilström curve but again the advantage is if you know your particle diameter you just go up here and you say ok if I have this velocity at that velocity the material will start eroding so it's easy to use this advantage like I said it's not dimensionless this graph is valid for a layer of water on top of the sediment of 100 centimeters if you have a different thickness of the layer of water you get a different graph so for each water layer you need a different graph because it's not dimensionless but everybody is using this graph and I never saw in literature a set of graphs for different water depths so everybody is ignoring the fact that this is for one meter but it is this is what you get if you project the shield curves in the Jilström graph and if you look carefully you can see here 1, 2, 3, 4 graphs I did it for 1 centimeter water depth 0.11 and 10 meters of water depth and then you can see how those curves differ then the red lines here are for here sand and here clay the green lines are for sand so if you look at the green line and you have pure sand in fact very fine sand we call silt you get those green lines so it continues going down but those lines are exactly the shields curves for those water depths this is a graph which has different names it's also often called the Parker shields graph it's in the coordinates of the shields graph here you have the Reynolds number in this case it's the particle Reynolds number which is a little bit different from the boundary Reynolds number in the lecture notes I give you all the equations to go from one Reynolds number to the other so you can make the transition tau star is the shields number is the dimensionless shear stress so basically this is the shields curve and this line here is the shields curve in this graph they use the so called Brownlee equations so Mr. Brownlee is one of the people who made fit functions for the shields curve so the black line is the Brownlee equation now what can you do with this graph and that's something I will use for the exam as well if you for a certain situation if you calculate where you are in this graph you can say ok if I'm below the black line no erosion if I'm above the black line so we forget the bandwidth that we have if I'm above the black line I have erosion but the graph tells you much more if I'm above the red line the material will be in suspension so all the particles are floating if I'm below the red line the particles are what we call saltating saltating means a particle when it starts moving moves up in the flow and because of gravity it falls down again so it's like jumping of particles so that's what you have below the red line dan we have this purple line if I'm on the left side of the purple line I have ripples and ripples are for example if you go to the beach and at low tide you see all those small ripples on the beach so ripples are vertical motions of the sediment but with a small amplitude gewoon ripples in Dutch if I'm on the right side of the purple line I have dunes so in fact you could say dunes are ripples with a much higher amplitude basically that's what it is and it's also a little bit nonsense to say there is one line dividing the two because in nature what will happen if I go from left to right the amplitude will slowly increase and then if I make a definition for example below 1 cm amplitude I call it a ripple and above I call it a dune then that's the division but in reality it's a gradual increase of the amplitude then if you are in this area you don't have dunes you have a flat bed the person who made this graph of course he based all those lines on many experiments it's never black and white so you can't say in one area it's always flat there will be some researcher in this area who also measured ripples or dunes that's possible but on average that's what you get so if you have to make a prediction of what kind of flow do I expect I can use this graph why is it important to know where you are well the resistance the lambda value that we get from the moody diagram depends on what we call the relative roughness of the surface well if I have flow through a steel pipe the relative roughness is almost zero I call it a smooth pipe but if I have sediment and on top of the sediment I have ripples of let's say 1 centimeter then suddenly the roughness of my bed is 1 centimeter it's not the particle diameter anymore it's the height of the ripples if I have dunes then the height of the dunes is the roughness of my bed and that's what I should use in the moody diagram to calculate the friction factor but that's why it's important to know something about what are we talking about so if I look at flow through a pipeline then I can actually see that if I'm transporting an 0.1 millimeter cent that means I'm all the way in this range it will be in suspension and most probably I have some small ripples so I can calculate the laptop value of my bed but if I would have gravel through that pipe and I'm in this range I could have much higher ripples which we call dunes and that means the resistance factor over there is much higher and I should know those things because the dredging contractor would like to know how much power do I need to pump the material through the pipeline ja een little bit about the flow if you look at the flow and in this case we look at flow in a river or a canal or the top of the hopper so this is not a closed system like a pipeline but an open system then we assume that above the bed we have a viscous layer because there the velocities are relatively small we assume that we have a laminar flow not no turbulence, laminar flow and in fact in reality the thickness of that layer is smaller than what you see in this picture but I needed to get this text in between so I had to increase the thickness of the layer in the picture so those thicknesses are not reality they are a little bit exaggerated well then you get a transition layer en then the turbulent logarithmic layer so this is the viscous sub-layer and this is the turbulent layer and that could be important because if I have very small particles which are entirely in the viscous sub-layer then the flow around those particles is completely different from particles that are so big that they are in the turbulent layer the CD value you already saw the CD value the CD value for laminar flow the stokes region could become very high while the CD value in the turbulent layer for big particles is just 0.445 or something near there so you get a completely different behavior whether it's in the viscous sub-layer or in the turbulent layer and in between you always have a transition I'm not gonna read this you can follow this on blackboard because I put the whole presentation on blackboard again then something we use constantly not just here but also in some other subjects friction velocity in this subject they use the term friction velocity to determine in fact the shear stress on the bed friction velocity is not a velocity they call it friction velocity but in reality it's not a velocity but it has the dimension of velocity so how do you use it friction velocity u star is the square root of the shear stress on the bed divided by the density of the fluid that's the definition even if you do a dimensional analysis you will find that it's the dimension of velocity and that's why they call it friction velocity but remember it's not a velocity so there is not a specific point in the flow where you can say that's where I have my friction velocity there will be a point where the velocity is equal to the friction velocity but it's not defined as a point in the flow where you have this velocity well, how can I determine it because if you look at the first equation you will say, yeah, very nice but still I don't know the shear stress and if I don't know the shear stress how do I know my friction velocity well, there is a general equation u star squared is lambda divided by 8 the lambda is the lambda that you get from the moody diagram I will explain after this and the u squared is the average velocity above the bed so if I know, for example in this hopper I still have 1 meter of water above the bed and I know my flow in cubic meters per second I can calculate the average velocity above the bed and that's my u the reason why I put critical here in fact I should remove it because that's to be exactly on the shield's curve you have a u critical if you are on the shield but to use this equation forget the critical it's just the average velocity above the bed so if I know this velocity and I know it and in a river it would be the average flow velocity in the river if I know it all I need to know is my lambda but for the lambda I need to know a little bit more well, here we have one of the equations for lambda you can see two equations the only difference is here I use the natural logarithm and here the ten logarithm nowadays everybody will say we use the natural logarithm but in the past before we had computers wat did people do? we had this double logarithmic paper and that was based on the ten logarithm so people would put all their measurements on that kind of paper try to make a straight line through those points and then you always get equations based on the ten logarithm so it's just because we had that kind of paper and that's what we used en er is no philosophical background behind it so the only difference is those two logarithms and you can see that it only results in a different factor in the nominator that's all so if you would calculate the ln of ten you have that factor that's between those two so what do we have in the equation let's take this one d normally is the diameter of the particles of the sediment if I assume a smooth surface then I can use the diameter of the particles and the big d is the hydraulic diameter of the flow so in a pipe it would be the diameter of the pipe but if you have a rectangular cross section you use the hydraulic diameter and next time so that's tomorrow I will go into detail about the diameter because like we just discussed you have a hydraulic diameter and a hydraulic radius and for mechanical engineers everybody will say so the hydraulic radius is half the hydraulic diameter it's not unfortunately it's one fourth of the diameter doesn't make sense to mechanical engineers but the reason for it is that in civil engineering, in hydraulic engineering you like to have a number that matches the water depth of a river or canal because you often do calculations on that and if you take one fourth of the hydraulic diameter you have the water depth in the case where you have a very wide river and not too much water depth I will show you that tomorrow ok, so this is the diameter of the particles diameter of the hydraulic diameter and this is the Reynolds number of the flow so it's not the Reynolds number of the particle or anything it's just the average velocity of the flow both times the hydraulic diameter divided by the viscosity in case you know you have ripples with a certain height for this d, for this small d you could use the height of those ripples if you know them normally I just work with the diameter of the particles but it depends on what you know about the situation if you don't know, you shouldn't just start using something here we see two cases so you see the velocity distribution again above the bed here you have the viscous sub-layer and here you have the turbulent layer here you see a particle or an obstacle which is entirely inside the viscous sub-layer so if I determine flow forces on that object I have to use the velocity distribution in the viscous sub-layer but here you can see particles that are bigger than the viscous sub-layer and then I also have to take into account the turbulent flow now in the viscous sub-layer this is your velocity distribution so this is the velocity this is the friction velocity this is the distance above the bed and this is the viscosity so you can see if my distance above the bed is zero my velocity is zero and that's the no slip condition of flow and you can see that it goes with the square of the friction velocity but since this is a constant in a certain case this gives me a linear velocity distribution in the turbulent layer this is the equation so you get a logarithmic profile where z zero is some constant depending on the type of flow that you have z again is the distance above the bed and here you have the friction velocity and kappa is the I thought the Boltzmann constant I know the von Karman constant which is roughly 0.4 so that's how you in the Boltzmann equation you also use it ok then we have a classification of flows you have smooth flow and hydraulically smooth flow is when this is a Reynolds number in fact this is the Reynolds number Mr. Shields was using where the ks is the roughness of the bed and if you have a smooth bed you take the diameter of the particle but if it's not smooth you have to take well often in civil engineering they use five times the diameter I think you have numbers for that rough flow is when this Reynolds number is above 70 so this is below 5 this is above 70 and in between you have so called transitional flow but transitional flow is the equations you will find is always just some mathematics to make two curves match there's no real theory behind it this in fact shows those velocity distributions so here you see the linear equation and here you see the logarithmic equation we already saw hydraulically smooth below 5 for this Reynolds number and as long as you don't know more just assume ks is the particle diameter it's roughly in the order the magnitude of the particle diameter so if I have smooth flow my z0 for the turbulent layer is this equation if I have rough flow it's proportional to the particle diameter and in between you get a combination of the two in the transition range you don't have to remember all those things first of all because the examination is open book and you can take all those slides or the lecture notes but one thing you should know is this one this is the thickness of the viscous sublayer that's an important one it's 11.6 times the viscosity divided by the u star the friction velocity I found in currently in my research for hydraulic transport so transport of slurry through a pipeline that this thickness is very important in what exactly is happening in the pipeline if I have a higher viscosity I get a thicker layer and the behavior in the pipeline is different in fact in that hydraulic transport one thing they also use is maybe you can imagine that if you have that viscous sublayer normally it's fluid flow nice linear velocity distribution but what happens if I have a lot of very fine particles inside that viscous sublayer then those particles make it more difficult for the fluid to slide over each other because in fact in the viscous sublayer you start at the sediment with velocity zero and then each level of water is flowing a little bit faster so you get layers of water sliding over each other but if there are particles they obstruct this sliding of the layers of water effectively that means the viscosity of the fluid is increasing and they have equations for that and for example if I would have an 0.1 millimeter cent 40% concentration then the viscosity increases by a factor of 12 based on certain equations that means suddenly my viscous sublayer is 12 times as thick because my viscosity is increasing that means that much bigger particles are still 100% in the viscous sublayer and that's what is actually happening and you can see that in many measurements well not directly but you can explain things that you measure by increasing the viscosity ok that's it for today and I thought I would finish the whole subject today but probably I talk too much you say yes